Problem: Let $f(x) = -x^{2}+5x-1$. Where does this function intersect the x-axis (i.e. what are the roots or zeroes of $f(x)$ )?
The function intersects the x-axis when $f(x) = 0$ , so you need to solve the equation: $-x^{2}+5x-1 = 0$ Use the quadratic formula to solve $ax^2 + bx + c = 0$ $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ $a = -1, b = 5, c = -1$ $ x = \dfrac{-5 \pm \sqrt{5^{2} - 4 \cdot -1 \cdot -1}}{2 \cdot -1}$ $ x = \dfrac{-5 \pm \sqrt{21}}{-2}$ $ x = \dfrac{-5 \pm \sqrt{21}}{-2}$ $x =\dfrac{-5 \pm \sqrt{21}}{-2}$